Small Subgraphs Research Articles

Detecting a planted community in an inhomogeneous random graph

We study the problem of detecting whether an inhomogeneous random graph contains a planted community. Specifically, we observe a single realization of a graph. Under the null hypothesis, this graph is a sample from an inhomogeneous random graph, whereas under the alternative, there exists a small subgraph where the edge probabilities are increased by a multiplicative scaling factor. We present a scan test that is able to detect the presence of such a planted community, even when this community is very small and the underlying graph is inhomogeneous. We also derive an information theoretic lower bound for this problem which shows that in some regimes the scan test is almost asymptotically optimal. We illustrate our results through examples and numerical experiments.

Subgraphs of Interest Social Networks for Diffusion Dynamics Prediction.

Finding the building blocks of real-world networks contributes to the understanding of their formation process and related dynamical processes, which is related to prediction and control tasks. We explore different types of social networks, demonstrating high structural variability, and aim to extract and see their minimal building blocks, which are able to reproduce supergraph structural and dynamical properties, so as to be appropriate for diffusion prediction for the whole graph on the base of its small subgraph. For this purpose, we determine topological and functional formal criteria and explore sampling techniques. Using the method that provides the best correspondence to both criteria, we explore the building blocks of interest networks. The best sampling method allows one to extract subgraphs of optimal 30 nodes, which reproduce path lengths, clustering, and degree particularities of an initial graph. The extracted subgraphs are different for the considered interest networks, and provide interesting material for the global dynamics exploration on the mesoscale base.

Fake review and reviewer detection through behavioral graph partitioning integrating deep neural network

With a profound effect of online reviews on customers’ decisions about purchasing products or services, untruthful (fake) reviews written to deceive product quality and receive unfair commercial benefits have become a crucial problem. In this work, we propose a graph partitioning approach (BeGP) and its extension (BeGPX) to distinguish fake reviewers from benign ones. The main idea of BeGP is to first construct a behavioral graph in which reviewers are connected if they share common characteristic features that capture their similar behavior. Then, the algorithm starts with a small subgraph of known fake reviewers and afterwards repeatedly expands the subgraph by inducing other connected suspicious reviewers. Subsequently, all reviews of those suspects are hypothesized to be untruthful. Moreover, to enhance the performance of fake review(er) detection, BeGPX employs additional analysis of semantic content and emotions expressed in reviews. In particular, we use the deep neural network to learn word embeddings representation and lexicon-based emotion indicators in order to integrate into the graph construction process. We demonstrate the effectiveness of BeGP and BeGPX on two real-world review datasets from Yelp.com. The results show that both approaches outperform state-of-the-art methods with accurately identifying fake review(er)s within the k-first order of rankings. In addition, BeGPX shows significant enhancement although being provided with only a few amount of learning labeled data.

Atomic subgraphs and the statistical mechanics of networks.

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges, but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real-world networks. More specifically, we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distributions of multiple atomic subgraphs that enables the construction of models with fewer parameters. Expanding the model to include atoms with edge and vertex labels we obtain a general class of models that can be parametrized in terms of basic building blocks and their distributions that include many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs, random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree-corrected and directed versions. We show that the entropy for all these models can be derived from a single expression that is characterized by the symmetry groups of atomic subgraphs.

Learning Knowledge Using Frequent Subgraph Mining from Ontology Graph Data

In many areas, vast amounts of information are rapidly accumulating in the form of ontology-based knowledge graphs, and the use of information in these forms of knowledge graphs is becoming increasingly important. This study proposes a novel method for efficiently learning frequent subgraphs (i.e., knowledge) from ontology-based graph data. An ontology-based large-scale graph is decomposed into small unit subgraphs, which are used as the unit to calculate the frequency of the subgraph. The frequent subgraphs are extracted through candidate generation and chunking processes. To verify the usefulness of the extracted frequent subgraphs, the methodology was applied to movie rating prediction. Using the frequent subgraphs as user profiles, the graph similarity between the rating graph and new item graph was calculated to predict the rating. The MovieLens dataset was used for the experiment, and a comparison showed that the proposed method outperformed other widely used recommendation methods. This study is meaningful in that it proposed an efficient method for extracting frequent subgraphs while maintaining semantic information and considering scalability in large-scale graphs. Furthermore, the proposed method can provide results that include semantic information to serve as a logical basis for rating prediction or recommendation, which existing methods are unable to provide.

Partition-Merge: Distributed Inference and Modularity Optimization

This paper presents a novel meta-algorithm, Partition-Merge (PM), which takes existing centralized algorithms for graph computation and makes them distributed and faster. In a nutshell, PM divides the graph into small subgraphs using our novel randomized partitioning scheme, runs the centralized algorithm on each partition separately, and then stitches the resulting solutions to produce a global solution. We demonstrate the efficiency of the PM algorithm on two popular problems: computation of Maximum A Posteriori (MAP) assignment in an arbitrary pairwise Markov Random Field (MRF) and modularity optimization for community detection. We show that the resulting distributed algorithms for these problems become fast, which run in time linear in the number of nodes in the graph. Furthermore, PM leads to performance comparable – or even better – to that of the centralized algorithms as long as the graph has polynomial growth property. More precisely, if the centralized algorithm is a $\mathcal {C}-$ factor approximation with constant $\mathcal {C}\ge 1$ , the resulting distributed algorithm is a $(\mathcal {C}+\delta)$ -factor approximation for any small $\delta >0$ ; and even if the centralized algorithm is a non-constant (e.g., logarithmic) factor approximation, then the resulting distributed algorithm becomes a constant factor approximation. For general graphs, we compute explicit bounds on the loss of performance of the resulting distributed algorithm with respect to the centralized algorithm. To show the efficiency of our algorithm, we conducted extensive experiments both on real-world networks and on synthetic networks. The experiments demonstrate that the PM algorithm provides a good trade-off between accuracy and running time.

Adversarial Attack on Large Scale Graph

Recent studies have shown that graph neural networks (GNNs) are vulnerable against perturbations due to lack of robustness and can therefore be easily fooled. Currently, most works on attacking GNNs are mainly using gradient information to guide the attack and achieve outstanding performance. However, the high complexity of time and space makes them unmanageable for large scale graphs and becomes the major bottleneck that prevents the practical usage. We argue that the main reason is that they have to use the whole graph for attacks, resulting in the increasing time and space complexity as the data scale grows. In this work, we propose an efficient Simplified Gradient-based Attack (SGA) method to bridge this gap. SGA can cause the GNNs to misclassify specific target nodes through a multi-stage attack framework, which needs only a much smaller subgraph. In addition, we present a practical metric named Degree Assortativity Change (DAC) to measure the impacts of adversarial attacks on graph data. We evaluate our attack method on four real-world graph networks by attacking several commonly used GNNs. The experimental results demonstrate that SGA can achieve significant time and memory efficiency improvements while maintaining competitive attack performance compared to state-of-art attack techniques.

Complete immersions in graphs with independence number two and small forbidden subgraphs

The analogue of Hadwiger’s conjecture for the immersion order states that every graph G contains the complete graph KX(G) as an immersion. Like its minor-order counterpart it is open even for graphs with independence number 2. Let G and H be graphs with independence number at most 2, such that |V(H)| ≤ 4. We show that if G is H-free, then G satisfies the conjecture.

An Approximate Blow-up Lemma for Sparse Hypergraphs

We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs.

Nonlinear Least Absolute Value Estimator for Topology Error Detection and Robust State Estimation

Topology error, a modeling misrepresentation of the power system network configuration, can undermine the quality of state estimation. In this paper, we propose a new methodology for robust power system state estimation (PSSE) modeled by AC power flow equations when there exists a small number of topological errors. The developed technique utilizes the availability of a large number of SCADA measurements and minimizes the $\ell _{1}$ norm of nonconvex residuals augmented by a nonlinear, but convex, regularizer. Representing the power network by a graph, we first study the properties of the solution obtained from the proposed NLAV estimator and demonstrate that, under mild conditions, this solution identifies a small subgraph of the network that contains the topological errors in the model used for the state estimation problem. Then, we introduce a method that can efficiently detect the topological errors by searching over the identified subgraph. In addition, we develop a theoretical upper bound on the state estimation error to guarantee the accuracy of the proposed state estimation technique. The efficacy of the developed framework is demonstrated through numerical simulations on IEEE benchmark systems.

Temporal Network Motifs: Models, Limitations, Evaluation

Investigating the frequency and distribution of small subgraphs with a few nodes/edges, i.e., motifs, is an effective analysis method for static networks. Motif-driven analysis is also useful for temporal networks where the spectrum of motifs is significantly larger due to the additional temporal information on edges. This variety makes it challenging to design a temporal motif model that can consider all aspects of temporality. In the literature, previous works have introduced various models that handle different characteristics. In this work, we compare the existing temporal motif models and evaluate the facets of temporal networks that are overlooked in the literature. We first survey four temporal motif models and highlight their differences. Then, we evaluate the advantages and limitations of these models with respect to the temporal inducedness and timing constraints. In addition, we suggest a new lens, event pairs, to investigate temporal correlations. We believe that our comparative survey and extensive evaluation will catalyze the research on temporal network motif models.

Graphlets importance ranking in complex networks based on the spectral energy contribution

ABSTRACT Recursive decomposition of networks is a widely used approach in network analysis for factorization of network structure into small subgraph patterns with few nodes. These patterns are called graphlets (motifs), and their analysis is considered as a common approach in bioinformatics. This paper focuses on evaluating the importance of graphlets in networks and proposes a new analytical model for ranking the graphlets importance based on their contribution to the graph energy spectrum. Besides, a general formula is provided to calculate the graphlet energy contribution to the total energy of a graph; then the energy value of the graph is estimated based on its graphlets. The results of the empirical analysis of synthetic and real networks are consistent with the theoretical results and suggest that the proposed analytical model can accurately estimate the structural features of a given graph based on its graphlets.

Learning Clique Subgraphs in Structural Brain Network Classification with Application to Crystallized Cognition

Structural brain networks constructed from diffusion MRI are important biomarkers for understanding human brain structure and its relation to cognitive functioning. There is increasing interest in learning differences in structural brain networks between groups of subjects in neuroimaging studies, leading to a variable selection problem in network classification. Traditional methods often use independent edgewise tests or unstructured generalized linear model (GLM) with regularization on vectorized networks to select edges distinguishing the groups, which ignore the network structure and make the results hard to interpret. In this paper, we develop a symmetric bilinear logistic regression (SBLR) with elastic-net penalty to identify a set of small clique subgraphs in network classification. Clique subgraphs, consisting of all the interconnections among a subset of brain regions, have appealing neurological interpretations as they may correspond to some anatomical circuits in the brain related to the outcome. We apply this method to study differences in the structural connectome between adolescents with high and low crystallized cognitive ability, using the crystallized cognition composite score, picture vocabulary and oral reading recognition tests from NIH Toolbox. A few clique subgraphs containing several small sets of brain regions are identified between different levels of functioning, indicating their importance in crystallized cognition.

Covering Small Subgraphs of (Hyper)Tournaments with Spanning Acyclic Subgraphs

While the edges of every tournament can be covered with two spanning acyclic subgraphs, this is not so if we set out to cover all acyclic $H$-subgraphs of a tournament with spanning acyclic subgraphs, even for very simple $H$ such as the $2$-edge directed path or the $2$-edge out-star. We prove new bounds for the minimum number of elements in such coverings and for some $H$ our bounds determine the exact order of magnitude.
 A $k$-tournament is an orientation of the complete $k$-graph, where each $k$-set is given a total order (so tournaments are $2$-tournaments). As opposed to tournaments, already covering the edges of a $3$-tournament with the minimum number of spanning acyclic subhypergraphs is a nontrivial problem. We prove a new lower bound for this problem which asymptotically matches the known lower bound of covering all ordered triples of a set.

Maximum common property: a new approach for molecular similarity

The maximum common property similarity (MCPhd) method is presented using descriptors as a new approach to determine the similarity between two chemical compounds or molecular graphs. This method uses the concept of maximum common property arising from the concept of maximum common substructure and is based on the electrotopographic state index for atoms. A new algorithm to quantify the similarity values of chemical structures based on the presented maximum common property concept is also developed in this paper. To verify the validity of this approach, the similarity of a sample of compounds with antimalarial activity is calculated and compared with the results obtained by four different similarity methods: the small molecule subgraph detector (SMSD), molecular fingerprint based (OBabel_FP2), ISIDA descriptors and shape-feature similarity (SHAFTS). The results obtained by the MCPhd method differ significantly from those obtained by the compared methods, improving the quantification of the similarity. A major advantage of the proposed method is that it helps to understand the analogy or proximity between physicochemical properties of the molecular fragments or subgraphs compared with the biological response or biological activity. In this new approach, more than one property can be potentially used. The method can be considered a hybrid procedure because it combines descriptor and the fragment approaches.

Link Prediction through Deep Generative Model.

SummaryInferring missing links based on the currently observed network is known as link prediction, which has tremendous real-world applications in biomedicine, e-commerce, social media, and criminal intelligence. Numerous methods have been proposed to solve the link prediction problem. Yet, many of these methods are designed for undirected networks only and based on domain-specific heuristics. Here we developed a new link prediction method based on deep generative models, which does not rely on any domain-specific heuristic and works for general undirected or directed complex networks. Our key idea is to represent the adjacency matrix of a network as an image and then learn hierarchical feature representations of the image by training a deep generative model. Those features correspond to structural patterns in the network at different scales, from small subgraphs to mesoscopic communities. When applied to various real-world networks from different domains, our method shows overall superior performance against existing methods.

Accurate, efficient and scalable training of Graph Neural Networks

Graph Neural Networks (GNNs) are powerful deep learning models to generate node embeddings on graphs. When applying deep GNNs on large graphs, it is still challenging to perform training in an efficient and scalable way. We propose a novel parallel training framework. Through sampling small subgraphs as minibatches, we reduce training workload by orders of magnitude compared with state-of-the-art minibatch methods. We then parallelize the key computation steps on tightly-coupled shared memory systems. For graph sampling, we exploit parallelism within and across sampler instances, and propose an efficient data structure supporting concurrent accesses from samplers. The parallel sampler theoretically achieves near-linear speedup with respect to number of processing units. For feature propagation within subgraphs, we improve cache utilization and reduce DRAM traffic by data partitioning. Our partitioning is a 2-approximation strategy for minimizing the communication cost compared to the optimal. We further develop a runtime scheduler to reorder the training operations and adjust the minibatch subgraphs to improve parallel performance. Finally, we generalize the above parallelization strategies to support multiple types of GNN models and graph samplers. The proposed training outperforms the state-of-the-art in scalability, efficiency and accuracy simultaneously. On a 40-core Xeon platform, we achieve 60× speedup (with AVX) in the sampling step and 20× speedup in the feature propagation step, compared to the serial implementation. Our algorithm enables fast training of deeper GNNs, as demonstrated by orders of magnitude speedup compared to the Tensorflow implementation. We open-source our code at https://github.com/GraphSAINT/GraphSAINT

Seeded graph matching via large neighborhood statistics

We study a noisy graph isomorphism problem, where the goal is to perfectly recover the vertex correspondence between two edge‐correlated graphs, with an initial seed set of correctly matched vertex pairs revealed as side information. We show that it is possible to achieve the information‐theoretic limit of graph sparsity in time polynomial in the number of vertices n. Moreover, we show the number of seeds needed for perfect recovery in polynomial‐time can be as low as in the sparse graph regime (with the average degree smaller than ) and in the dense graph regime, for a small positive constant . Unlike previous work on graph matching, which used small neighborhoods or small subgraphs with a logarithmic number of vertices in order to match vertices, our algorithms match vertices if their large neighborhoods have a significant overlap in the number of seeds.

Continuous Monitoring of Maximum Clique Over Dynamic Graphs

The maximum clique problem ( <i>MCP</i> ) has various applications to reveal the structure and function of graphs. Graphs are constantly updated in the real life. However, no algorithm is specifically designed for dynamic graph. Although <i>MCP</i> in dynamic graphs can be solved by simply invoking a state-of-the-art static approach, such as <i>PMC</i> , when the graph is updated, such an approach of simply re-calculating from scratch is inefficient. The key issue with <i>MCP</i> algorithm is to find a large clique, namely a <i>seed</i> , as fast as possible. Thus, search space can be pruned based on the seed. Size of the seed greedily found by <i>PMC</i> cannot be guaranteed, as it fluctuates considerably. Moreover, the time required to find a seed under <i>PMC</i> is up to <inline-formula><tex-math notation="LaTeX">$O(| E| \cdot \Delta (G))$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\Delta (G)$</tex-math></inline-formula> is the highest degree in <i>G</i> . In this article, we intend to find a sizable seed by updating the previous maximum clique with the incident vertices of the inserted/deleted edge. Size of the seed now is guaranteed to be no less than <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})\; - \;1$</tex-math></inline-formula> , where <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> is the size of the maximum clique on the updated graph. Moreover, the seed can be found in a time complexity of <inline-formula><tex-math notation="LaTeX">$O(\Delta (G)^{2})$</tex-math></inline-formula> . Two other crucial issues related to the <i>MCP</i> in dynamic graphs are refreshing rate and refreshing overhead. After a tight upper bound is imposed on <inline-formula><tex-math notation="LaTeX">$\omega (G^{\prime})$</tex-math></inline-formula> , the necessity of refreshing is evaluated by comparing the seed with its largest challenger, then unnecessary refreshing is wiped out effectively. The size of the largest challenger is judiciously estimated using a lazy growth strategy. Subsequently, the search space in refreshing is confined on a much smaller subgraph using a local refreshing strategy. Extensive experiments indicate that the proposed approach outperforms the baseline algorithm by approximately one order of magnitude.

Processing Grid-format Real-world Graphs on DRAM-based FPGA Accelerators with Application-specific Caching Mechanisms

Graph processing is one of the important research topics in the big-data era. To build a general framework for graph processing by using a DRAM-based FPGA board with deep memory hierarchy, one of the reasonable methods is to partition a given big graph into multiple small subgraphs, represent the graph with a two-dimensional grid, and then process the subgraphs one after another to divide and conquer the whole problem. Such a method (grid-graph processing) stores the graph data in the off-chip memory devices (e.g., on-board or host DRAM) that have large storage capacities but relatively small bandwidths, and processes individual small subgraphs one after another by using the on-chip memory devices (e.g., FFs, BRAM, and URAM) that have small storage capacities but superior random access performances. However, directly exchanging graph (vertex and edge) data between the processing units in FPGA chip with slow off-chip DRAMs during grid-graph processing leads to limited performances and excessive data transmission amounts between the FPGA chip and off-chip memory devices. In this article, we show that it is effective in improving the performance of grid-graph processing on DRAM-based FPGA hardware accelerators by leveraging the flexibility and programmability of FPGAs to build application-specific caching mechanisms, which bridge the performance gaps between on-chip and off-chip memory devices, and reduce the data transmission amounts by exploiting the localities on data accessing. We design two application-specific caching mechanisms (i.e., vertex caching and edge caching ) to exploit two types of localities (i.e., vertex locality and subgraph locality ) that exist in grid-graph processing, respectively. Experimental results show that with the vertex caching mechanism, our system (named as FabGraph) achieves up to 3.1× and 2.5× speedups for BFS and PageRank, respectively, over ForeGraph when processing medium graphs stored in the on-board DRAM. With the edge caching mechanism, the extension of FabGraph (named as FabGraph+) achieves up to 9.96× speedups for BFS over FPGP when processing large graphs stored in the host DRAM.